SPACE-TIME : ARENA OR ILLUSION

68 Pages·1999·2.52 MB·English

SLAC - PUB - 3156
July 1983
T/E
SPACE-TIME : ARENA OR ILLUSION ?*
VADIM KAPLUNCVSKY+ and MARVIN WEINSTEIN
Stanford Linear Accelerator Center
Stanford University, Stanford, California, 94 305
ABSTRACT
a-
- This paper develops a framework which allows us to treat the topology and
dimension of the space-time continuum as dynamically generated. We present
examples of quantum systems which are defined without a notion of space, but
which nevertheless undergo a transition to a space-time phase. The dimension
of the space is an integer valued order parameter which characterizes distinct
phases of a single system. We also show the interactions between the low energy
particles of the system are gauge-like. Finally, we discuss the computability of
Newton’s constant in this class of theories.
Submitted to Physics Review D
* Work supported by the Department of Energy, contract D E - AC 0 3 - 7 6 S F 00 5 15
+ Address after September 1, 1983: Department of Physics, Princeton University, Princeton,
N. J. 08544
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1. INTRODUCTION
Self-consistency problems in many areas of physics have forced theorists to
try to achieve a better understanding of phenomena at, or near, the Planck
scale. Such considerations, whether they have to do with the structure of the
early universe and the flatness problemf or with applications of supergravity to
solving the hierarchy problemf necessitate that we learn to incorporate Einstein
gravity into the structure of quantum field theory. Unfortunately, to date, no
satisfactory way of doing this has been proposed.
Many talented physicists have failed to solve the problem of renormalizing a
conventional field theory of gravity plus matter, and the concommitant problem
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of “fine tuning” the cosmological constant. For this reason we feel the time is ripe
for a reappraisal of basic tenets with an eye towards eliminating unnecessary as-
sumptions. This paper presents the result of one such reappraisal. We find there
is a way of formulating quantum field theory which violates our most cherished
beliefs and yet appears capable of describing physics as we know it. The notion
abandoned as superfluous to a quantum field theory is that of the four dimen-
sional space-time continuum. There seem to be quantum systems which start
out with a well-defined notion of time but no notion of space, and dynamically
undergo a transition to a space-time phase - a phase in which the physics of the
low energy degrees of freedom of the system are best described by an effective
Lagrangian written in terms of conventional relativistic fields. In this sense, the
notion of the four dimensional space-time continuum as the arena within which
the game of field theory is to be played is replaced by the notion of the space-time
continuum as an illusion of low-energy dynamics.
On the face of it the idea that one can abandon relativity and the space-
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time continuum as superfluous baggage seems ridiculous. For this reason we
begin with a discussion of some of the ideas which led us to this approach. We
go on to describe the class of Hamiltonians suggested by these considerations,
and present the methods we will use to analyze such systems. In particular, we
anticipate results of the sections to follow and discuss the notions of dimension
and scattering for a theory formulated without reference to an underlying space-
time. The next section introduces simple examples and argues that for these
examples low energy physics is correctly described by effective lfl-dimensional
relativistic field theories. This is followed by a section devoted to the problem of
obtaining higher dimensional space-times. We present toy Hamiltonians which
-I- lead to effective 2 + 1 and 3 + 1 relativistic space-times, thus demonstrating that
this problem is solvable. We also discuss the relationship between the space-
time dmensions, internal “flavor” symmetries, and of possible phases of a given
Hamiltonian. The last section summarizes results and presents a list of caveats
and speculations about those aspects of the problem not yet studied in detail.
2. FUNDAMENTALS
2.1 MACH'S PRINCIPLE AND THE COSMOLOGICAL CONSTANT.
A suggestive way of looking at the problem of “fine tuning the cosmological
constant” is that it might be related to the fact that the Einstein theory does
not successfully embody Mach’s principle: which asserts that the local inertial
properties of matter should be determined by the other (in particular distant)
matter in the universe. This assertion implies that in a really empty universe
a “test particle” should not know how to move; as a corollary, an empty space
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should have no geometry at all. On the other hand, the Einstein equations allow
for a Aat space geometry
in a universe devoid of matter and radiation.
This failure of Einstein’s theory to incorporate Mach’s principle is likely to
be related to the problem of “fine tuning” the cosmological constant, which arises
when one attempts to couple quantum matter fields to general relativity. Quan-
-I- turn fields generally have a non-zero vacuum energy, whose contribution to the
energy-momentum tensor is equivalent to an enormous cosmological constant.
Unless this contribution is precisely cancelled by an equally enormous bare cos-
mological constant, i.e. “fine tuned away”, Einstein’s space-time curls up into
a tightly curved manifold. In contrast to this situation, a theory which obeys
Mach’s principle and yields no geometry at all for a truly empty space, i.e. de-
tll
void of even quantum fluctuations, might yield a flat geometry when quantum
effects are taken into account. In this case the flatness problem would have a
natural solution. However, now we have to solve another problem: How to arrive
at such a desirable state of affairs?
As a starting point we focused upon the fact that our ideas of space-time are
derived by carrying out measurements with clocks and rulers, the latter being
replaced by radar sets in more modern expositions of the subject! In other
words, our notions of space and time are the result of carrying out a large class
tfl We mean, a flat geometry for the background space-time metric of the low-energy physics.
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of scattering experiments. The necessary input to this constructive approach is
the existence of some objects that can be scattered and a Hamiltonian which
describes the scattering process. Space-time, if it is an appropriate description of
low energy physics, then emerges as a construct. This sort of argument naturally
leads us to inquire as to what role the coordinates x,, play in conventional field
theories.
2.2 WHAT ARE THE x,'s?
Space-time is usually treated as an arena for dynamical theories, even when
one takes into account the Einstein approach to gravity as geometry. To better
2- appreciate this fact, let us reexamine the usual formulation of quantum field
theory. The Hamiltonian for a self-interacting scalar field in m-dimensions is
written as
H= I12(x) + pq2 (x) + v (Q(z))] * (2-l)
What is the significance of the variables 5,’ which appear in this formula?
At the level of the canonical commutation relations the xP’s are nothing but labels
for independent operators Q(x) and II(x) which define the theory. A priori there
is no reason for xP’s to have anything to do with space and time. However, the
way the gradient terms appear in (2.1) ensures that space-time defined by this
Hamiltonian leads us back to the same xP’s (up to a Lorentz transformation).
This rather non-trivial result implies that a preconceived notion of the topological
and diflerentiable structure is associated with the underlying variables and that
certain aspects of xP’s must be taken into account before we define the quantum
problem. In fact, great care must be taken in renormalizing the theory so as not
to do great violence to this relationship.
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At first glance Einstein’s theory takes care of these troubles. In this case we
consider an action of the form
L = dmx e [ R(X) + matter terms ] , (24
/
which is invariant with respect to arbitrary coordinate transformations of the
variables xP. This invariance seems to reduce the significance of the xP’s to their
role as labels for independent quantum degrees of freedom. From a heuristic
point of view this approach has another beautiful property to recommend it: if
the background metric of space-time arises as a dynamical effect, then there is
2-
a natural reason why the space-time reconstructed from scattering experiments
must come out intimately related to the variables xP. The equations of general
relativity, and therefore the Heisenberg equations of the quantum theory (assum-
ing that the theory can be quantized) form a set of hyperbolic partial differential
equations. It is known5 that for this set of differential equations the causal
hypersurfaces of the combined system of fields are determined by the causal hy-
persurfaces of the universally coupled spin-two field, i.e. gPv . ( By the causal
surfaces of the fields, we mean the submanifolds of the space-time across which
the fields may be discontinuous, i.e. the light cones of the theory. ) It follows
that if gPV develops a ground state expectation value, iPV, then solving the quan-
tum problem for the matter fields will reconstruct a theory whose light cones are
those specified by iPV. If in particular ijPV = qllV, the space-time constructed
by scattering the matter fields will be the desired Minkowski space. This seems
to be a highly attractive scenario, modulo difficulties in defining the quantum
theory; unfortunately this heuristic picture isn’t completely correct.
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Although going over to the Einstein theory of gravity removes the geometri-
cal significance of the coordinates xcL, the fact that they specify the differential
topology of the system remains. In other words, although the notion of dis-
tance has become a dynamical concept, we still need to introduce the notions
of ‘neighborhood’ and ‘deriuatioe’ as primitive, i.e. undefined, terms. When we
assume the xP’s to be coordinate patches on a certain manifold, we establish
a certain topological and differentiable structure for the -space-time. Since the
primitive notion of differentiability must be defined before we specify an action
or a Hamiltonian, the assumption that we are dealing with a theory defined on
a differentiable manifold carries non-trivial content.
.- -
2.3 BASICS.
Motivated by these considerations, we explore the question of whether formu-
lating a theory so that the analogs of the xP’s are reduced to labels for quantum
degrees of freedom6 will allow us to progress towards the goal of incorporating
Mach’s principle into field theory. We begin by introducing an indexing set
J = (l:l=l,...,N) .
We then introduce a set of quantum “harmonic oscillator”, or “bose field”, vari-
ables II(Z) and Q(Z) , satisfying canonical commutation relations
and multicomponent “Fermion field” variables \E, (1) and X4!/(:1 ) , satisfying canon-
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ical anti-commutation relations
Intuitively, these quantum degrees of freedom are to be thought of as playing the
role of site variables in some sort of lattice field theory.
In addition to these site fields we also introduce link j&Z&, which for the sake
of definiteness we choose to be harmonic oscillator variables Pap and XaP
Um) (lm) ’ The
notation ‘(Zm)’ is introduced to stand for an arbitrary pair of integers I, m E J,
ordered such that 1 < m. These link fields are assumed to satisfy the canonical
.-- commutation relations
The role of the link fields is to provide dynamical variables which allow us to
write the analogue of a kinetic term for a Hamiltonian without requiring that the
notion of differentiation or even neighborhood to be defined.
Starting with these objects we consider a Hamiltonian of the form
H = c [$ H”(Z) + Xfm@(I) a(m) + V(Q(Z))
- iXl,a { Q+(Z) K@(m) - ‘E+(m) Q(Z) }
+ f P”r,, + v (- * - X(1,) * * *) ] .
Note, to simplify the discussion we will temporarily treat the fermion fields as
one component objects and drop the Dirac indices CYa nd /3. We also introduce a
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matrix notation for the variables X(lm):
1X( 1,) for 1 < m ,
Xl, = - Xml = (2.4)
Xtrnl) for m < 1 ;
and define X”, to be
Xfm = c &k Xkm - (2.5)
kcJ
As it stands, (2.3) is the Hamiltonian of a complicated quantum mechanical
system all of whose couplings, for want of a better choice, are assumed to be of
order unity. The idea we wish to pursue is that, for some dynamical reason, a
subset of the variables X(1,) acquire non-vanishing ground state (i.e. vacuum)
.--
expectation values X(lrn) . When this occurs, the terms quadratic in the fields
6(l) and K@(Zy) ield a solvable zeroth order Hamiltonian for the “matter fields” in
_ the system and, to the extent that fluctuations in the fields X(1,) can be ignored,
map the matter problem into some sort of lattice theory. The terms involving
variables X(lm) which have non-vanishing expectation values are identifiable as
the derivative term in this effective lattice Hamiltonian. Two main questions
are: “What drives the dynamical generation of expectation values of the fields
X(lm) ?“, and “What sort of lattice forms?“. While the meaning of the first
question is self-evident, the second merits a brief discussion.
If the ground-state of a Hamiltonian of the form defined in (2.3) corresponds
to the vanishing vacuum expectation value of the fields Xtlrn) , no zeroth order
geometry (or effective lattice theory) - reasonable or unreasonable - forms. In
other words there is no sense in which the matter fields scatter in a background
determined by the vacuum expectation value of the link fields; everybody inter-
acts on an equal footing with everyone else. On the other hand, if a pattern
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of vacuum expectation values does develop, then there is a zeroth order matter
field theory whose arena of definition is specified by those link variables with
non-vanishing expectation values. Our problem is to determine if there is any
reasonably simple way of choosing terms for (2.3) so that the resulting theory
becomes equivalent to a Hamiltonian lattice theory with dynamics closely ap-
proximating that of a relativistic continuum theory. The aim of this paper is
to show that for a large class of Hamiltonians of the type specified in (2.3), the
second possibility is realized. In addition, we will show that simply by increasing
the number of degrees of freedom the resulting lattice theory can be made to
approximate a relativistic continuum theory arbitrarily well.
.a-
2.4 BISONS VERSUS FERMIONS AS FUNDAMENTAL BUILDING BLOCKS
Before discussing specific calculations, let us see if we can intuitively un-
derstand those aspects of the theory which control the dynamical generation of
ground-state expectation values for link-variables. Obviously the potential terms,
can determine whether the theory develops non-zero expecta-
v(*‘*X(lm) *.‘) 9
tion values of the link variables; however the vacuum energy of the matter fields
can play a similar role. The possibility that matter fields cause spontaneous
generation of non-vanishing link-field expectation values by themselves, i.e. irre-
spective of the details of the link Hamiltonian, is very appealing. In a moment
we will argue this possibility is realized if the dominant matter fields in the sys-
tem are fermionic. Unfortunately we will find that fermionic fields alone cannot
stabilize the resulting space-time structure against quantum fluctuations of the
link variables. This will lead us to consider generalized plaquette terms in the
--
link-field potential (we will define them later), which can produce stable space-
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t